An efficient second-order scalar auxiliary variable approach for unsteady non-Newtonian incompressible fluids

Abstract

A novel second-order time-splitting method is proposed for the numerical solution of non-Newtonian fluid flows governed by the incompressible Navier-Stokes equations with a shear-rate dependent viscosity. In many applications, this class of fluid flows is challenging to numerically solve using the conventional monolithic methods. The proposed approach belongs to a family of viscosity-splitting methods and it separates the convection term from the incompressibility constraint into two steps using the second-order implicit backward differentiation formula for the time integration. To ensure the stability of this method, we introduce a numerical scheme based on the scalar auxiliary variable approach an efficient pressure incrementation is introduced using the scalar auxiliary variable approach. Unlike most projection methods for solving incompressible Navier-Stokes equations, the proposed method is free of any numerical inconsistencies generated by the treatment of boundary conditions in the pressure solution. A rigorous stability analysis is also carried out in this study and the proposed method is demonstrated to be consistent and stable with no restrictions on the time step. Numerical results are presented for three flow problems to validate the second-order convergence rates and to illustrate the performance of the proposed time-splitting scheme for unsteady non-Newtonian incompressible fluids.